$P$ be a plane in $\Bbb{R}^3$ containing origin, $v:[0,1]\to P$ be an integrable vector valued map. Is $\int_{0}^{1}v(t) dt\in P $ true?

Question

If I formalize the question into linear algebra terms it will look like this-
Here P is a $2$ dimensional vector subspace of $\Bbb{R}^3$.
Construct $V=\{\alpha:[0,1]\to\Bbb{R}^3|\alpha$ is integrable$\}$. It is vector spcae with respect to point wise addition and pointwise scalar multiplication.
Construct $T:V\to\Bbb{R}^3$, $T(\alpha)=\int_{0}^{1}\alpha(t) dt$. This is linear map. Let $S=\{\alpha\in V|\alpha(t)\in P\ \forall t\in[0,1]\}$.
Then my question will be - is it true that $T(S)\subseteq P$?
I don't know how to proceed further. Can anybody give an idea to answer the question? Thanks for assistance in advance.

Answer 1

The answer is positive and the consequence that for finite dimensional spaces, the integral of a vectorial map is equal to the vector having for coordinates the integrals of each coordinate. You then just have to consider a basis of $P$.

You can have a look at the properties of Bochner integral.

Answer 2

The plane $P$ is a subspace of ${\mathbb R}^3$. There is a unit vector ${\bf n}\perp P$, and $${\bf x}\in P\quad\Leftrightarrow\quad{\bf n}\cdot{\bf x}=0\ .$$ If $\alpha(t)\in P$ for all $t$ then ${\bf n}\cdot \alpha(t)=0$ for all $t$, and $${\bf n}\cdot T(\alpha)=\int_0^1{\bf n}\cdot\alpha(t)\>dt=0\ ,$$ hence $T(\alpha)\in P$.